Gyrokinetic Microtearing Studies

Transcription

1 1 Gyrokinetic Microtearing Studies C M Roach summarising work involving many collaborators: D J Applegate 0, JW Connor, S C Cowley, D Dickinson 1, W Dorland 2, R J Hastie, S Saarelma, A A Schekochihin 3 and H R Wilson 1 Euratom/CCFE Fusion Association, Culham Science Centre, UK 0 SERCO 1 University of York, UK 2 University of Maryland, US 3 University of Oxford, UK This work was funded by the United Kingdom Engineering and Physical Sciences Research Council and the European Communities under the contract of Association between EURATOM and CCFE. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

2 2 Outline of the Talk Summary of mostly old gyrokinetic simulations of microtearing modes in STs, using GS2. (1) Tearing Parity Modes and Simulation Literature (2) Microtearing Mode in MAST (3) Contact with Analytic Theory (4) Nonlinear Simulations (5) Key Questions

3 3 Eigen-Mode Parity along Equilibrium Magnetic Field is Even or Odd Local ballooning space represents physical quantities as twisting slices: ( θ θ ) ( ) inq ( x )2 p θ θ 2π p e ik y ( y + s x ) 0 π F ( x, y, θ ) = e F 0 fast variation p = slow ll variation ˆ x is equ m flux surface label, x=0 at q(x)=m/n y equ m field line label, to b, lying in the flux surface θ is ll to b Fˆ is defined on infinite domain in the ballooning angle η, θ 0 is the ballooning parameter. ˆF ( η ) 0 asη ± Fˆ eigenfunctions are either even or odd in η, about η= θ 0

4 4 Tearing Parity Modes Fˆ ( η ) At x=0, the parity of about η= θ 0 in ballooning space determines the symmetry of F along the field line in real space Perturbed magnetic field comes from δb = δa radial component:δb x = A ll / y = ik y A ll A ll even, conclude for x=0 that δb x same sign along equ m field line δb x sinusoidal in y at fixed θ equilibrium field lines are torn! Even A implies tearing of magnetic flux surface x=0

5 5 Some Gyrokinetic Microtearing Mode Simulations in the Literature Microtearing found in study high β and high performance plasmas: M Kotschenreuther et al, Nuclear Fusion 40, 677 (2000) GS2 Often dominant instabilities for k y ρ i < 1 at mid-radius in MAST plasmas: D J Applegate et al, Phys Plasmas 11, 5085 (2004) C M Roach et al, PPCF 47, B323 (2005) GS2 Microtearing found to dominate ST Power Plant equilibrium: H R Wilson et al, Nuclear Fusion 44, 917 (2004) GS2 Detailed numerical study of microtearing, ST reference, includes scan in R/a: D J Applegate et al, PPCF 49, 1113 (2007) GS2 Nonlinear analytic theory of µ-tearing may explain electron transport in NSTX K L Wong et al, Phys. Rev. Lett. 99, (2007) Edge plasmas in ASDEX-Upgrade have µ-tearing modes D Told et al, Phys. Plasmas 15, (2008) GENE!

6 6 Linear Microstability Analysis at Mid-Radius in MAST MAST equilibrium from ELMy H-Mode #6252 At mid-radius surface Ψ n =0.4, β e =0.05, q~1.35 T i ~ T e, a~0.3m, R~0.9m R/a~3 See Applegate et al, Physics of Plasmas (2004)

7 7 Tearing Parity Modes at ρ i scale Fastest growing modes in STs often found to have tearing parity: MAST [1], and NSTX [2] conceptual burning STs [3,4] [1] Applegate et al, Phys Plasmas 11, 5085, (2004). [2] Redi et al, EPS, St Petersburg (2003) [3] Kotschenreuther et al, Nuc Fus 40, 677 (2000), [4] H R Wilson et al, Nuclear Fusion, 44, 917 (2004) MAST tearing parity modes rotate in electron diamagnetic drift direction

8 8 Visualising Micro-tearing Mode in Real Space Poincaré plot shows perturbed magnetic field at intersection of GS2 flux-tube with the outboard mid-plane. y (ρi ) Magnetic island on rational surface at x=0. Microtearing mode is candidate to explain electron transport Two Major Questions: x(ρ i ) What is the linear physics mechanism underlying these modes? How much anomalous transport is generated at nonlinear saturation?

9 9 Analytic Theories of Microtearing Instabilities T e microtearing drive discovered in cylinder Hazeltine Dobrott and Wang (1975): kinetic, collisions key, any ν e /ω Further slab calculations confirm T e drive at high ν e /ω Drake and Lee (1977), Gladd et al (1980): kinetic, Hassam (1980): fluid => collisional slab drive requires energy dependent ν e (E) Kinetic calculations in toroidal geometry (large R/a), for low ν e /ω Catto and Rosenbluth (1981), Connor, Cowley and Hastie (1990) low collisionality drive from trapped particle collisions on passing particles also requires energy dependent ν e (E) MAST MAST has small R/a and ν e /ω ~0.5 so analytic theories should be poor. Catto-Rosenbluth trapped particle drive mechanism, nevertheless, predicts growth with MAST parameters!...connor, Cowley, Hastie does not! γ/ω (C-R) ν e /ω CM Roach et al, PPCF 47, B323 (2005)

10 10 Analytic Theories of Microtearing Drives and Properties of the GS2 Modes Two classes of linear drive in analytic theory literature: time dependent thermal force (high collisionality, υ ei >ω) collisions close to the trapped-passing boundary (υ ei <ω) Both drives require finite dt e /dr energy dependent collision frequency ν ei (v) Some properties of the GS2 mode: sensitive to electron physics ν e, T e and n e sensitive to β, p, s insensitive to ion parameters ν i and T i and δb current layer width ~O(ρ i ) [1] DJ Applegate et al, PPCF 49, 1113 (2007) and PhD Imperial College (2006)

11 11 Experiment with Collision Operator DJ Applegate et al, PPCF 49, 1113 (2007) and PhD Imperial College (2006) GS2 Lorentz collision operator can capture boundary layers. Removed energy dependent collisions by setting ν e (E)=constant + E dependentυ ei E independentυ ei Modest affect on tearing γ not consistent with analytic drive models!

12 12 Experiments Using s-α Model Equilibrium: Scan Aspect Ratio by varying R 0 at Fixed r DJ Applegate et al, PPCF 49, 1113 (2007) Fit MAST mid-radius surface with s-α model for fixed β, a/l T, a/l n, q, s Scan r/r 0 by varying R 0 and fixing r and other parameters, varies drifts + f t MAST equ m r/r 0 = r/r 0 =0.236 r/r 0 =0.118 MAST instability in s- α too shaping not essential γ as r/r 0 still unstable at r/r 0 =0.118 µtearing may appear at conventional aspect ratio

13 13 Experiments Using s-α Model Equilibrium: Scan R 0 at fixed r/r 0 to Vary Drifts DJ Applegate et al, PPCF 49, 1113 (2007) Now scan in R 0 at fixed r/r 0 with other parameters constant R 0 = R 0 = 1.32 R 0 = 2.64 R 0 = Mode survives at R 0 = i.e. zero drift mode has slab drive

14 14 Experiments Using s-α Model Equilibrium: Scan in Trapped Particle Fraction, f t DJ Applegate et al, PPCF 49, 1113 (2007) Now scan r/r 0 to vary f t at fixed R 0 and other parameters r/r 0 = r/r 0 = r/r 0 = r/r 0 = 0. f t High f t γ at low ν e γ at high ν e (fewer passing e) Low f t γmore sensitive to energy dependent collision rate ν e (E)

15 15 Overview of Most Interesting Findings DJ Applegate et al, PPCF 49, 1113 (2007) Microtearing mode is driven by dt e /dr as expected. Mode is complicated and in awkward regime for analytic theory: unstable over broad range of collisionality 0.05<υ ei /ω<1.2 current layer width ~ O(ρ i ), so need ion FLR effects Regimes where mode robust to energy independent collisions puzzle Mode not only unstable in ST unstable in large aspect ratio s-α model equilibria Gyrokinetic microtearing also at r/r ~ 0.3 (~ MAST mid-radius) in conventional aspect ratio: D Told et al, Phys. Plasmas 15, (2008)

16 16 * Very High β: Microstability in STPP see H R Wilson et al, Nuc Fus 44, 917 (2004) Conceptual Culham ST Power Plant (STPP), 1GW electrical, β=0.59 GS2 used for microstability analysis of mid-radius flux-surface, Ψ n =0.35. Equilibrium features: striking variation in B around the magnetic flux surface magnetic drift reversal owing to high pressure gradient diamagnetic ω se strongly peaked on outboard midplane

17 17 * Microstability Results for Mid-radius Surface in STPP STPP surface Ψ n =0.35 no electrostatic instabilities, α stabilisation giving drift reversal including EM gives tearing parity modes at ion and electron scales k y ρ I =0.4 k y ρ i =6

18 18 * Microstability Results for Mid-radius Surface in STPP STPP surface Ψ n =0.35 no electrostatic instabilities (α stabilisation from drift reversal) EM effects gives tearing parity modes at ion and electron scales, all propagating in electron drift direction Mixing length χ~4m 2 s -1 (no ω se ) GK electrons + adiabatic ions, less extended along B =>cheaper! 4 species + highly extended along B => expensive!

19 19 Nonlinear Microtearing Simulations with GS2 D J Applegate First nonlinear GK simulations with GS2 [1,2] : modified mid-radius MAST equilibrium for increased tractability reduces radial box size by factor 5 Few k y modes: n ky =4, n kx =47, n θ =32 n E =8, n λ ~20 pseudo-saturation with low transport, blows up later at high k x small timesteps imposed by the CFL condition [1] D J Applegate PhD Thesis, Imperial College (2007). [2] D J Applegate et al, 32nd EPS, Tarragona, ECA volume 29C, P5-101, 2005

20 20 Impact of Adding Dissipation at High k R J Akers et al, IAEA FEC, Geneva, October 2008 EX/2-2 Use hyperviscosity for high k dissipation, parameterised by D no impact on linear physics improves convergence saturation insensitive to D but what are we throwing away? D= 0, 0.5, 1, 2

21 21 Nonlinear Electron Heat Flux D J Applegate Hyperviscosity smoothes high k x spike events reappear at nky=8 A contribution dominates q e low heat fluxes at saturation k y ρ i = 0, 0.2, 0.4, 0.6, D=0 k y ρ i = 0, 0.2, 0.4, 0.6, D=1 k y ρ i = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, D=1

22 22 Poincaré Plot and δj contours at θ=0 D J Applegate before spike event, t=532

23 23 Poincaré Plot and δj contours at θ=0, t=598. D J Applegate after spike event perturbed field wanders further, transport

24 24 A Spectra for nky=8 Simulation Spikes most evident at high k, but are controlled by D highest k middle k lowest finite k x, or k y =0 steady growth in zonal modes

25 25 * Φ Spectra for nky=8 Simulation Spikes most evident at high k, but suppressed by D highest k middle k lowest finite k x, or k y =0

26 26 Fidelity Issues Convergence? saturation sensitive to Min(k y ), and we need to go lower in k y! what causes the high k spikes? are we dissipating important physics? Flux-Tube equilibrium? as reduce Min(k y ρ i ), we go to low n s SIM = 5 s MAST so L x artificially small at lower k y and s, flux-tube gets fatter, to challenge local approximations More work needed! D J Applegate

27 27 Do Microtearing Modes Matter in MAST Anyway? D Dickinson, York Impact of FLOW SHEAR on microtearing modes? γ E > γ lin so will they be suppressed? slab drive may make suppression more difficult almost done

28 28 Conclusions Microtearing modes from GS2 simulations of MAST are complicated! trapped and passing particles contribute drive with dt e /dr insensitivity of γ to energy dependent collision frequency is puzzling µtearing specific neither to ST geometry nor to GS2! linear benchmark? map out where µtearing important Limited comparisons with analytic theory so far. do better in easier limits? Preliminary nonlinear simulations for MAST mid-radius are interesting, but: more work needed to test convergence what is happening at high k? local flux-tube equilibrium is challenged if n gets too small! easier equilibria? impact of FLOW SHEAR?